You cannot connect to modern servers. You need to find a dedicated "Alpha 1.0.16_02" server or host your own using an old server JAR (available from the official Mojang version manifest archive).
This version is famous for three things:
Follow these instructions precisely to experience Alpha 1.0.16_02 without risking your computer or Mojang account.
The render distance is fixed at "Short" by modern standards, but it's masked by an incredible atmospheric fog. You cannot see the edge of the world; it just fades into a solid color. Caves are pitch black. Without a dynamic light mod (which didn't exist yet), torches create a harsh circle of light.
So open your launcher, scroll past "Latest Release," and step into the fog. Just remember to bring a stack of torches. The dark is much, much darker back there.